In combinatorial mathematics, the Lobb number Lm, n counts the number of ways that n + m open parentheses can be arranged to form the start of a valid sequence of balanced parentheses.
The Lobb number are parameterized by two non-negative integers m and n with n >= m >= 0. It can be obtained by:
Lobb Number is also used to count the number of ways in which n + m copies of the value +1 and n – m copies of the value -1 may be arranged into a sequence such that all of the partial sums of the sequence are non- negative.
Examples :
Input : n = 3, m = 2 Output : 5 Input : n =5, m =3 Output :35
The idea is simple, we use a function that computes binomial coefficients for given values. Using this function and above formula, we can compute Lobb numbers.
// CPP Program to find Ln, m Lobb Number. #include <bits/stdc++.h> #define MAXN 109 using namespace std;
// Returns value of Binomial Coefficient C(n, k) int binomialCoeff( int n, int k)
{ int C[n + 1][k + 1];
// Calculate value of Binomial Coefficient in
// bottom up manner
for ( int i = 0; i <= n; i++) {
for ( int j = 0; j <= min(i, k); j++) {
// Base Cases
if (j == 0 || j == i)
C[i][j] = 1;
// Calculate value using previously stored values
else
C[i][j] = C[i - 1][j - 1] + C[i - 1][j];
}
}
return C[n][k];
} // Return the Lm, n Lobb Number. int lobb( int n, int m)
{ return ((2 * m + 1) * binomialCoeff(2 * n, m + n)) / (m + n + 1);
} // Driven Program int main()
{ int n = 5, m = 3;
cout << lobb(n, m) << endl;
return 0;
} |
// JAVA Code For Lobb Number import java.util.*;
class GFG {
// Returns value of Binomial
// Coefficient C(n, k)
static int binomialCoeff( int n, int k)
{
int C[][] = new int [n + 1 ][k + 1 ];
// Calculate value of Binomial
// Coefficient in bottom up manner
for ( int i = 0 ; i <= n; i++) {
for ( int j = 0 ; j <= Math.min(i, k);
j++) {
// Base Cases
if (j == 0 || j == i)
C[i][j] = 1 ;
// Calculate value using
// previously stored values
else
C[i][j] = C[i - 1 ][j - 1 ] +
C[i - 1 ][j];
}
}
return C[n][k];
}
// Return the Lm, n Lobb Number.
static int lobb( int n, int m)
{
return (( 2 * m + 1 ) * binomialCoeff( 2 * n, m + n)) /
(m + n + 1 );
}
/* Driver program to test above function */
public static void main(String[] args)
{
int n = 5 , m = 3 ;
System.out.println(lobb(n, m));
}
} // This code is contributed by Arnav Kr. Mandal. |
# Python 3 Program to find Ln, # m Lobb Number. # Returns value of Binomial # Coefficient C(n, k) def binomialCoeff(n, k):
C = [[ 0 for j in range (k + 1 )]
for i in range (n + 1 )]
# Calculate value of Binomial
# Coefficient in bottom up manner
for i in range ( 0 , n + 1 ):
for j in range ( 0 , min (i, k) + 1 ):
# Base Cases
if (j = = 0 or j = = i):
C[i][j] = 1
# Calculate value using
# previously stored values
else :
C[i][j] = (C[i - 1 ][j - 1 ]
+ C[i - 1 ][j])
return C[n][k]
# Return the Lm, n Lobb Number. def lobb(n, m):
return ((( 2 * m + 1 ) *
binomialCoeff( 2 * n, m + n))
/ (m + n + 1 ))
# Driven Program n = 5
m = 3
print ( int (lobb(n, m)))
# This code is contributed by # Smitha Dinesh Semwal |
// C# Code For Lobb Number using System;
class GFG {
// Returns value of Binomial
// Coefficient C(n, k)
static int binomialCoeff( int n, int k)
{
int [, ] C = new int [n + 1, k + 1];
// Calculate value of Binomial
// Coefficient in bottom up manner
for ( int i = 0; i <= n; i++) {
for ( int j = 0; j <= Math.Min(i, k);
j++) {
// Base Cases
if (j == 0 || j == i)
C[i, j] = 1;
// Calculate value using
// previously stored values
else
C[i, j] = C[i - 1, j - 1]
+ C[i - 1, j];
}
}
return C[n, k];
}
// Return the Lm, n Lobb Number.
static int lobb( int n, int m)
{
return ((2 * m + 1) * binomialCoeff(
2 * n, m + n)) / (m + n + 1);
}
/* Driver program to test above function */
public static void Main()
{
int n = 5, m = 3;
Console.WriteLine(lobb(n, m));
}
} // This code is contributed by vt_m. |
<?php // PHP Program to find Ln, // m Lobb Number. $MAXN =109;
// Returns value of Binomial // Coefficient C(n, k) function binomialCoeff( $n , $k )
{ $C = array ( array ());
// Calculate value of Binomial
// Coefficient in bottom up manner
for ( $i = 0; $i <= $n ; $i ++)
{
for ( $j = 0; $j <= min( $i , $k ); $j ++)
{
// Base Cases
if ( $j == 0 || $j == $i )
$C [ $i ][ $j ] = 1;
// Calculate value using p
// reviously stored values
else
$C [ $i ][ $j ] = $C [ $i - 1][ $j - 1] +
$C [ $i - 1][ $j ];
}
}
return $C [ $n ][ $k ];
} // Return the Lm, n Lobb Number. function lobb( $n , int $m )
{ return ((2 * $m + 1) *
binomialCoeff(2 * $n , $m + $n )) /
( $m + $n + 1);
} // Driven Code $n = 5; $m = 3;
echo lobb( $n , $m );
// This code is contributed by anuj_67. ?> |
<script> // javascript code for Lobb Number // Returns value of Binomial
// Coefficient C(n, k)
function binomialCoeff(n, k)
{
let C = new Array(n + 1);
// Loop to create 2D array using 1D array
for ( var i = 0; i < C.length; i++) {
C[i] = new Array(2);
}
// Calculate value of Binomial
// Coefficient in bottom up manner
for (let i = 0; i <= n; i++) {
for (let j = 0; j <= Math.min(i, k);
j++) {
// Base Cases
if (j == 0 || j == i)
C[i][j] = 1;
// Calculate value using
// previously stored values
else
C[i][j] = C[i - 1][j - 1] +
C[i - 1][j];
}
}
return C[n][k];
}
// Return the Lm, n Lobb Number.
function lobb(n, m)
{
return ((2 * m + 1) * binomialCoeff(2 * n, m + n)) /
(m + n + 1);
}
// Driver code let n = 5, m = 3;
document.write(lobb(n, m));
// This code is contributed by sanjoy_62.
</script> |
Output
35
Time Complexity: O(2*n*(m+n))
Auxiliary Space: O((2*n)*(m+n))
Efficient approach: Space optimization
In previous approach the current value dp[i][j] is only depend upon the current and previous row values of DP. So to optimize the space complexity we use a single 1D array to store the computations.
Implementation steps:
- Create a 1D vector C of size K+1.
- Set a base case by initializing the values of C.
- Now iterate over subproblems by the help of nested loop and get the current value from previous computations.
- At last return and print the final answer stored in C[K].
Implementation:
// CPP Program to find Ln, m Lobb Number. #include <bits/stdc++.h> #define MAXN 109 using namespace std;
// Returns value of Binomial Coefficient C(n, k) int binomialCoeff( int n, int k)
{ int C[k+1];
memset (C, 0, sizeof (C));
C[0] = 1; // nC0 is 1
// Calculate value of Binomial Coefficient
for ( int i = 1; i <= n; i++)
{
for ( int j = min(i, k); j > 0; j--)
C[j] = C[j] + C[j-1];
}
//return final answer
return C[k];
} // Return the Lm, n Lobb Number. int lobb( int n, int m)
{ return ((2 * m + 1) * binomialCoeff(2 * n, m + n)) / (m + n + 1);
} // Driven Program int main()
{ int n = 5, m = 3;
// function call
cout << lobb(n, m) << endl;
return 0;
} |
import java.util.Arrays;
public class LobbNumber {
// Returns value of Binomial Coefficient C(n, k)
static int binomialCoeff( int n, int k) {
int [] C = new int [k + 1 ];
Arrays.fill(C, 0 );
C[ 0 ] = 1 ; // nC0 is 1
// Calculate value of Binomial Coefficient
for ( int i = 1 ; i <= n; i++) {
for ( int j = Math.min(i, k); j > 0 ; j--) {
C[j] = C[j] + C[j - 1 ];
}
}
//return final answer
return C[k];
}
// Return the Lm, n Lobb Number.
static int lobb( int n, int m) {
return (( 2 * m + 1 ) * binomialCoeff( 2 * n, m + n)) / (m + n + 1 );
}
// Driven Program
public static void main(String[] args) {
int n = 5 , m = 3 ;
// function call
System.out.println(lobb(n, m));
}
} |
# Returns value of Binomial Coefficient C(n, k) def binomialCoeff(n, k):
C = [ 0 ] * (k + 1 )
C[ 0 ] = 1 # nC0 is 1
# Calculate value of Binomial Coefficient
for i in range ( 1 , n + 1 ):
j = min (i, k)
while j > 0 :
C[j] = C[j] + C[j - 1 ]
j - = 1
# return final answer
return C[k]
# Return the Lm, n Lobb Number. def lobb(n, m):
return (( 2 * m + 1 ) * binomialCoeff( 2 * n, m + n)) / / (m + n + 1 )
# Driven Program if __name__ = = "__main__" :
n = 5
m = 3
# function call
print (lobb(n, m))
|
using System;
public class Program
{ // Returns value of Binomial Coefficient C(n, k)
static int binomialCoeff( int n, int k)
{
int [] C = new int [k + 1];
Array.Fill(C, 0);
C[0] = 1; // nC0 is 1
// Calculate value of Binomial Coefficient
for ( int i = 1; i <= n; i++)
{
for ( int j = Math.Min(i, k); j > 0; j--)
C[j] = C[j] + C[j - 1];
}
//return final answer
return C[k];
}
// Return the Lm, n Lobb Number.
static int lobb( int n, int m)
{
return ((2 * m + 1) * binomialCoeff(2 * n, m + n)) / (m + n + 1);
}
// Driven Program
public static void Main()
{
int n = 5, m = 3;
// function call
Console.WriteLine(lobb(n, m));
}
} |
function binomialCoeff(n, k) {
let C = new Array(k + 1).fill(0);
C[0] = 1; // nC0 is 1
// Calculate value of Binomial Coefficient for (let i = 1; i <= n; i++) {
for (let j = Math.min(i, k); j > 0; j--)
C[j] = C[j] + C[j - 1]; } //return final answer return C[k];
} function lobb(n, m) {
return ((2 * m + 1) * binomialCoeff(2 * n, m + n)) / (m + n + 1);
} // Driven Program let n = 5, m = 3; console.log(lobb(n, m)); |
Output
35
Time Complexity: O(n^2)
Auxiliary Space: O(k)